A New Method for Lower Bounds on the Running Time of Evolutionary Algorithms

TL;DR

This paper introduces a new, versatile method for establishing lower bounds on the expected running time of evolutionary algorithms, applicable to various functions and mutation rates, with implications for optimizing algorithm performance.

Contribution

The paper presents a novel method based on fitness-level partitions and transition probabilities, providing exact or near-exact lower bounds for a wide class of functions and algorithms.

Findings

Lower bounds applicable to all algorithms using bit-flip mutation.

Determination of optimal mutation rates for specific problems.

Implications for selecting the most efficient mutation-based algorithms.

Abstract

We present a new method for proving lower bounds on the expected running time of evolutionary algorithms. It is based on fitness-level partitions and an additional condition on transition probabilities between fitness levels. The method is versatile, intuitive, elegant, and very powerful. It yields exact or near-exact lower bounds for LO, OneMax, long k-paths, and all functions with a unique optimum. Most lower bounds are very general: they hold for all evolutionary algorithms that only use bit-flip mutation as variation operator---i.e. for all selection operators and population models. The lower bounds are stated with their dependence on the mutation rate. These results have very strong implications. They allow to determine the optimal mutation-based algorithm for LO and OneMax, i.e., which algorithm minimizes the expected number of fitness evaluations. This includes the choice of the optimal mutation rate.